Specific electrical resistivity
R_t or specific electrical conductivity
\sigma_t = \frac{1}{R_t} of formations is defined by mineralization of the rock matrix and saturating fluids which depend on water-saturated shaliness
V_{sh} , formation porosity
\phi_e and water saturation volumetric share
s_w.
Archie Model
Indonesia Model (Poupon-Leveaux)
Simandeux Model
Simandeux model suggest a more complicated correlation between resitvity R_t and water saturation s_w:
with default value A = 0.8.
Dual-Water Model (DW)
The dual-water model accounts for the fact that different shales have different shale-bound water saturation
s_{wb}= \frac{V_{wb}}{V_t}:
\phi_t = \phi_e + \phi_t s_{wb} |
so that formation water saturation s_w is related to total water saturation s_{wt} = \frac{V_{wb} + V_w}{V_t } as:
s_w = \frac{s_{wt} - s_{wb}}{ 1 - s_{wb}} |
Formation resistivity
R_t is given by the following correlation:
\frac{1}{R_t} = \phi_t^m s_{wt}^n \, \Big[ \frac{1}{R_w} + \frac{s_{wb}}{s_{wt}} \Big( \frac{1}{R_{wb}} - \frac{1}{R_w} \Big) \Big] \quad \Rightarrow \quad s_w = \frac{s_{wt} - s_{wb}}{ 1 - s_{wb}} |
where
s_{wb} = \frac{V_{wb}}{V_t} | shale-bound water saturation |
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s_{wt} = \frac{V_{wb} + V_w}{V_t} | total water saturation (shal-bound water and free-water) |
R_{wb} | specific electrical resisitvity of shale-bound water |
In simple case when all shales have the same properties, the shale-bound water saturation can be expressed through the shaliness as:
(1) | s_{wb} = \zeta_{wb} V_{sh} |